Closed convex hull of set of measurable functions, Riemann-measurable functions and measurability of translations

@article{Talagrand1982,
abstract = {Let $G$ be a locally compact group. Let $L_t$ be the left translation in $L^\infty (G)$, given by $L_tf(x) = f(tx)$. We characterize (undre a mild set-theoretical hypothesis) the functions $f\in L^\infty (G)$ such that the map $t\rightarrow L_tf$ from $G$ into $L^\infty (G)$ is scalarly measurable (i.e. for $\varphi \in L^\infty (G)^*$, $t\rightarrow \varphi (L_tf)$ is measurable). We show that it is the case when $t\rightarrow \theta (L_ft)$ is measurable for each character $\theta $, and if $G$ is compact, if and only if $f$ is Riemann-measurable. We show that $t \rightarrow L_tf$ is Borel measurable if and only if $f$ is left uniformly continuous.Some of the measure-theoretic tools used there have independent interest. For example, if a set of measurable functions on $[0,1]$ is separable and point-wise relatively compact, the same is true of its convex hull.},
author = {Talagrand, Michel},
journal = {Annales de l'institut Fourier},
keywords = {locally compact group; left translation; scalarly measurable; Borel measurable; left uniformly continuous},
language = {eng},
number = {1},
pages = {39-69},
publisher = {Association des Annales de l'Institut Fourier},
title = {Closed convex hull of set of measurable functions, Riemann-measurable functions and measurability of translations},
url = {http://eudml.org/doc/74530},
volume = {32},
year = {1982},
}

TY - JOUR
AU - Talagrand, Michel
TI - Closed convex hull of set of measurable functions, Riemann-measurable functions and measurability of translations
JO - Annales de l'institut Fourier
PY - 1982
PB - Association des Annales de l'Institut Fourier
VL - 32
IS - 1
SP - 39
EP - 69
AB - Let $G$ be a locally compact group. Let $L_t$ be the left translation in $L^\infty (G)$, given by $L_tf(x) = f(tx)$. We characterize (undre a mild set-theoretical hypothesis) the functions $f\in L^\infty (G)$ such that the map $t\rightarrow L_tf$ from $G$ into $L^\infty (G)$ is scalarly measurable (i.e. for $\varphi \in L^\infty (G)^*$, $t\rightarrow \varphi (L_tf)$ is measurable). We show that it is the case when $t\rightarrow \theta (L_ft)$ is measurable for each character $\theta $, and if $G$ is compact, if and only if $f$ is Riemann-measurable. We show that $t \rightarrow L_tf$ is Borel measurable if and only if $f$ is left uniformly continuous.Some of the measure-theoretic tools used there have independent interest. For example, if a set of measurable functions on $[0,1]$ is separable and point-wise relatively compact, the same is true of its convex hull.
LA - eng
KW - locally compact group; left translation; scalarly measurable; Borel measurable; left uniformly continuous
UR - http://eudml.org/doc/74530
ER -